AST-14

Luminosity distance (nearby)

d_L = (c / H₀) z (1 + z/2) in a matter-empty nearby expansion. Pedagogical.

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RadiationLuminosity distance

Governing equation

d_L\approx(c/H_0)z(1+z/2)

$z \in \left[0.001, 1.5\right]$

where

z: Redshift (—)
H_0: Hubble constant (km/s/Mpc)
d_L: Luminosity distance (Mpc)

The figure, part by part

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①

Governing law — red

d_L\approx(c/H_0)z(1+z/2)

$z \in \left[0.001, 1.5\right]$

$z = 0.001$ $d_L = 4.285$

Along the curve

②

Domain of the variable — blue

z_{\min} = 0.001,\quad z_{\max} = 1.5

$z \in \left[0.001, 1.5\right]$

$z = 0.001$ $z = 0.001$

Along the curve

③

Response curve — green

d_L\approx(c/H_0)z(1+z/2)

$z \in \left[0.001, 1.5\right]$

$z = 0.001$ $d_L = 4.285$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}d_L}{\mathrm{d}z}

$z \in \left[0.001, 1.5\right]$

$z = 0.01$ $d_L^{\prime} = 4327.4$

Along the curve

Lecture brief

Historical brief

Hubble expansion, Jeans collapse, Eddington luminosity, Bondi accretion and Stefan–Boltzmann stars are the first astrophysical budgets. The sheets scale a star, a cloud and a horizon. This sheet (AST-14 — Luminosity distance (nearby)) is the form associated with Luminosity distance. Working symbols: $z$ , $H_0$ $\rightarrow$ $d_L$ . At very low z, d_L ≈ c z / H₀. Flux F = L / (4π d_L²).

Purpose

Purpose: compute

d_L

from

z

H_0

in Astrophysics via

d_L\approx(c/H_0)z(1+z/2)

d_L = (c / H₀) z (1 + z/2) in a matter-empty nearby expansion. Pedagogical. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

z = 0.050\,\mathrm{—}

H_0 = 70.000\,\mathrm{km/s/Mpc}

, the governing relation

d_L\approx(c/H_0)z(1+z/2)

yields

d_L = 219.49\,\mathrm{Mpc}

. A candle, a stretch, a dimming. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Redshift z0.05 —Hubble constant H_070 km/s/Mpc

Outputs

Luminosity distance d_L219.49 Mpc

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Narration of this film

A candle, a stretch, a dimming.

At very low z, d_L ≈ c z / H₀. Flux F = L / (4π d_L²).

Reading speed

Watch on YouTube